How to animate the wrapping and unwrapping motion of rope around a pole like in the game TetherBall
You can use Soft Body physics to achieve this - similar to How to get the physics engine to wraps a string around the moving part of a motor constraint?
Here's a quick example :

The key here was to setup the 'string' as a string of vertices with Soft Body, setup the 'pole' with collision, and use a Wind force at the start of the animation (keyframed to just give an initial 'push' and then stop - with 'flow' set to zero) to give it the initial motion. The Soft Body simulation does the rest.